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Given a tree and a set ${\cal P}$ of non-trivial simple paths on it,$VPT({\cal P})$ is the VPT graph (i.e. the vertex intersection graph) of thepaths ${\cal P}$ of the tree $T$, and $EPT({\cal P})$ is the EPT graph (i.e.the edge intersection graph) of ${\cal P}$. These graphs have been extensivelystudied in the literature. Given two (edge) intersecting paths in a graph,their \emph{split vertices} is the set of vertices having degree at least $3$in their union. A pair of (edge) intersecting paths is termed\emph{non-splitting} if they do not have split vertices (namely if their unionis a path).  In this work, motivated by an application in all-optical networks, we definethe graph $ENPT({\cal P})$ of edge-intersecting non-splitting paths of a tree,termed the ENPT graph, as the (edge) graph having a vertex for each path in${\cal P}$, and an edge between every pair of paths that are bothedge-intersecting and non-splitting. A graph $G$ is an ENPT graph if there is atree $T$ and a set of paths ${\cal P}$ of $T$ such that $G=ENPT({\cal P})$, andwe say that $<T,{\cal P}>$ is a \emph{representation} of $G$. We first showthat cycles, trees and complete graphs are ENPT graphs.  Our work follows the lines of Golumbic and Jamison's research in which theydefined the EPT graph class, and characterized the representations of chordlesscycles (holes). It turns out that ENPT holes have a more complex structure thanEPT holes. In our analysis, we assume that the EPT graph corresponding to arepresentation of an ENPT hole is given. We also introduce three assumptions$(P1)$, $(P2)$, $(P3)$ defined on EPT, ENPT pairs of graphs. In this Part I,using the results of Golumbic and Jamison as building blocks, we characterize(a) EPT, ENPT pairs that satisfy $(P1)$, $(P2)$, $(P3)$, and (b) the uniqueminimal representation of such pairs.

Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Towards Hole Representations-Part I